Properties

Label 9405.2.a.w
Level $9405$
Weight $2$
Character orbit 9405.a
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(75.0993031010\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.131947641.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + 2) q^{4} - q^{5} + ( - \beta_{4} + \beta_1 + 1) q^{7} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + 2) q^{4} - q^{5} + ( - \beta_{4} + \beta_1 + 1) q^{7} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{8} + \beta_1 q^{10} + q^{11} + (\beta_{3} - \beta_1 - 2) q^{13} + ( - \beta_1 - 3) q^{14} + (\beta_{5} + \beta_{4} + \beta_{2} + \cdots + 1) q^{16}+ \cdots + (3 \beta_{4} + 3 \beta_1 - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7} + 6 q^{11} - 9 q^{13} - 18 q^{14} + 4 q^{16} + 5 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{23} + 6 q^{25} + 22 q^{26} + 10 q^{28} + 5 q^{29} - q^{31} - 15 q^{32} - 22 q^{34} - 5 q^{35} + 9 q^{37} - 25 q^{41} + 15 q^{43} + 8 q^{44} - 16 q^{46} - 24 q^{47} + 13 q^{49} - 27 q^{52} - 5 q^{53} - 6 q^{55} + 12 q^{56} + 13 q^{58} - 39 q^{59} - 11 q^{61} + 42 q^{62} - 14 q^{64} + 9 q^{65} + 24 q^{67} - 45 q^{68} + 18 q^{70} + 24 q^{71} - 26 q^{73} - q^{74} - 8 q^{76} + 5 q^{77} + 11 q^{79} - 4 q^{80} + 8 q^{82} - 39 q^{83} - 5 q^{85} - 18 q^{86} - 22 q^{89} - 26 q^{91} + 11 q^{92} - 30 q^{94} + 6 q^{95} + 22 q^{97} - 33 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 7\nu^{3} + 7\nu - 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 7\nu^{3} - 3\nu^{2} + 7\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 10\nu^{3} - 22\nu + 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{4} - 6\beta_{3} + 7\beta_{2} + \beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{4} + 10\beta_{2} + 28\beta _1 + 3 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56745
1.65636
0.759131
−0.577704
−2.04201
−2.36323
−2.56745 0 4.59179 −1.00000 0 2.16848 −6.65430 0 2.56745
1.2 −1.65636 0 0.743534 −1.00000 0 2.81120 2.08116 0 1.65636
1.3 −0.759131 0 −1.42372 −1.00000 0 4.95189 2.59905 0 0.759131
1.4 0.577704 0 −1.66626 −1.00000 0 −4.19297 −2.11801 0 −0.577704
1.5 2.04201 0 2.16980 −1.00000 0 −0.469142 0.346728 0 −2.04201
1.6 2.36323 0 3.58485 −1.00000 0 −0.269449 3.74537 0 −2.36323
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(11\) \( -1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9405.2.a.w 6
3.b odd 2 1 1045.2.a.g 6
15.d odd 2 1 5225.2.a.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.g 6 3.b odd 2 1
5225.2.a.k 6 15.d odd 2 1
9405.2.a.w 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):

\( T_{2}^{6} - 10T_{2}^{4} + 25T_{2}^{2} + 3T_{2} - 9 \) Copy content Toggle raw display
\( T_{7}^{6} - 5T_{7}^{5} - 15T_{7}^{4} + 90T_{7}^{3} - 55T_{7}^{2} - 81T_{7} - 16 \) Copy content Toggle raw display
\( T_{13}^{6} + 9T_{13}^{5} + 9T_{13}^{4} - 63T_{13}^{3} - 42T_{13}^{2} + 63T_{13} - 14 \) Copy content Toggle raw display
\( T_{17}^{6} - 5T_{17}^{5} - 51T_{17}^{4} + 189T_{17}^{3} + 936T_{17}^{2} - 1771T_{17} - 6290 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 10 T^{4} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 5 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 9 T^{5} + \cdots - 14 \) Copy content Toggle raw display
$17$ \( T^{6} - 5 T^{5} + \cdots - 6290 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 8 T^{5} + \cdots + 92 \) Copy content Toggle raw display
$29$ \( T^{6} - 5 T^{5} + \cdots + 482 \) Copy content Toggle raw display
$31$ \( T^{6} + T^{5} + \cdots - 3240 \) Copy content Toggle raw display
$37$ \( T^{6} - 9 T^{5} + \cdots + 1906 \) Copy content Toggle raw display
$41$ \( T^{6} + 25 T^{5} + \cdots + 9526 \) Copy content Toggle raw display
$43$ \( T^{6} - 15 T^{5} + \cdots + 4460 \) Copy content Toggle raw display
$47$ \( T^{6} + 24 T^{5} + \cdots - 7268 \) Copy content Toggle raw display
$53$ \( T^{6} + 5 T^{5} + \cdots - 627158 \) Copy content Toggle raw display
$59$ \( T^{6} + 39 T^{5} + \cdots + 35420 \) Copy content Toggle raw display
$61$ \( T^{6} + 11 T^{5} + \cdots - 156150 \) Copy content Toggle raw display
$67$ \( T^{6} - 24 T^{5} + \cdots - 123184 \) Copy content Toggle raw display
$71$ \( T^{6} - 24 T^{5} + \cdots - 56840 \) Copy content Toggle raw display
$73$ \( T^{6} + 26 T^{5} + \cdots + 34902 \) Copy content Toggle raw display
$79$ \( T^{6} - 11 T^{5} + \cdots + 8152 \) Copy content Toggle raw display
$83$ \( T^{6} + 39 T^{5} + \cdots + 2100044 \) Copy content Toggle raw display
$89$ \( T^{6} + 22 T^{5} + \cdots + 554 \) Copy content Toggle raw display
$97$ \( T^{6} - 22 T^{5} + \cdots + 506 \) Copy content Toggle raw display
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